Graph state representation of the toric code

Abstract

Given their potential for fault-tolerant operations, topological quantum states are currently the focus of intense activity. Of particular interest are topological quantum error correction codes, such as the surface and planar stabilizer codes that are equivalent to the celebrated toric code. While every stabilizer state maps to a graph state under local Clifford operations, the graphs associated with topological stabilizer codes remain unknown. We show that the toric code graph is composed of only two kinds of subgraphs: star graphs (which encode Greenberger-Horne-Zeilinger states) and half graphs. The topological order is identified with the existence of multiple star graphs, which reveals a connection between the repetition and toric codes. The graph structure readily yields a log-depth quantum circuit for state preparation, assuming geometrically non-local gates, which can be reduced to a constant depth including ancillae and measurements at the cost of increasing the circuit width. The results provide a new graph-theoretic framework for the investigation of topological order and the development of novel topological error correction codes.

arXiv comment18 pages, 8 figures. One section about encoding arbitrary state into toric code is added

References in corpus (30)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Discovery (theoretical prediction and experimental observation) of a large-gap topological-insulator class with spin-polarized single-Dirac-cone on the surface
• A generic new platform for topological quantum computation using semiconductor heterostructures
• Topological quantum memory
• Non-Abelian statistics and topological quantum information processing in 1D wire networks
• String-net condensation: A physical mechanism for topological phases
• Exponential Protection of Zero Modes in Majorana Islands
• Multi-party entanglement in graph states
• Stabilizer Codes and Quantum Error Correction
• Topological Quantum Distillation
• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• Topological quantum order: stability under local perturbations
• Universal topological quantum computation from a superconductor/Abelian quantum Hall heterostructure
• Graphical description of the action of local Clifford transformations on graph states
• A Short Introduction to Topological Quantum Computation
• Entanglement renormalization and topological order
• Repetition Cat Qubits for Fault-Tolerant Quantum Computation
• Efficient quantum algorithms for $GHZ$ and $W$ states, and implementation on the IBM quantum computer
• Adiabatic Preparation of Topological Order
• Foliated Quantum Codes
• Universal topological phase of 2D stabilizer codes
• Exact entanglement renormalization for string-net models
• Local permutations of products of Bell states and entanglement distillation
• Quantum circuit complexity of one-dimensional topological phases
• Braiding of Atomic Majorana Fermions in Wire Networks and Implementation of the Deutsch-Josza Algorithm
• Creating and manipulating a Laughlin-type $\nu =1/3$ fractional quantum Hall state on a quantum computer with linear depth circuits
• Weight of quadratic forms and graph states
• Weight Reduction for Quantum Codes
• General scheme for preparation of different topological states on the cluster states

Cited by in corpus (6)

• Topological graph states and quantum error correction codes
• Non-local finite-depth circuits for constructing SPT states and quantum cellular automata
• Representing arbitrary ground states of toric code by a restricted Boltzmann machine
• The additivity of states uniquely determined by marginals
• Stabilization of symmetry-protected long-range entanglement in stochastic quantum circuits
• Scaling of Computational Order Parameters in Rydberg Atom Graph States